Table of contents  
1. Superposition Theorem  
2. Thevenin's Theorem  
3. Norton's Theorem  
4. Maximum Power Transfer Theorem  
Frequently Asked Questions (FAQs) 
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The fundamental theory on which many branches of electrical engineering, 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. So, before jumping to the theorems let's first understand the conditions for Linear Networks.
An element is said to be linear if it satisfies homogeneity (scaling) property and additive (superposition) property.
Let x be the input and y be the output of an element.
If kx(t) is applied to the element, the output must be ky.
Showing homogeneity property
x_{1}(t) → y_{1(t)}, x_{2(t)} → y_{2(t)}
If (x_{1}(t) + x_{2}(t)) is applied to the element, the output must be y_{1}(t) + y_{2}(t).
If k(x_{1}(t) + x_{2}(t)) is applied to the element, the output must be k(y_{1}(t) + y_{2}(t)).
So, for a network to qualify the application of various theorems must follow the conditions given above.
Let's study some amazing theorems:
Procedure for using the superposition theorem
So for above given circuit the total response or say current I through register R_{2} will be equal to the sum of individual response obtained by each source.
I = I'_{due to E}_{1}_{(alone) }+ I''_{due to E2(alone) }+ I'''_{due to Ix(alone)}
Example: Here in the following electrical circuit, we will find the current flowing through the 10 Ω resistor using the superposition theorem.
Here at first let’s consider the 30 A current source. So we will leave the 30 A current source as it is in the circuit and replace the 60 V voltage source with the short circuit as shown below.
Now the current through 10 Ω resistor is calculated as
[The I_{1} is calculated using the current divider.]
Now let’s consider a 60 V voltage source. So we will leave the 60 V voltage source as it is and replace the 30 A current source with the open circuits shown below.
Then current through 10 Ω resistor is calculated as
Finally, the total current flowing through the 10 Ω resistor is the algebraic sum of I_{1} and I_{2}.
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.
Thevenin Circuit
The procedure for applying Thevenin’s theorem
Example: Find current flowing through 1 Ω resistor.
Open Load Resistor
Voltage Source are shorted
Equivalent Circuit to find R_{th}
Example: For the given circuit, determine the current flowing through 10 Ω resistor using Norton’s theorem.
Since the question here, is to determine the current through 10 Ω resistor, it is considered as the load.
(a) To find Norton’s current, Remove the load resistor(10 Ω), short it with a wire and the circuit is redrawn as below.
In this circuit, we need to find the current I_{N}, which is Norton’s current flowing from a to b. To find the value of I_{N}, it is necessary to determine the total current in the circuit.
If you observe the circuit, 3 Ω resistor and 2 Ω resistor are in parallel with each other. This parallel combination is in series with 1 Ω resistor. Thus,
Now, the total current I_{T} is given by,
The current through the 2 Ω resistor (or Norton’s current I_{N}) is obtained by applying current division rule:
(b) To find Norton’s resistance, remove the load resistor, short the voltage source and circuit is redrawn as below.
In this circuit, we can observe that the 2 Ω resistor is in series with the parallel combination of 1 Ω and 3 Ω resistors. Thus the equivalent value of resistance is obtained as,
(c) Norton’s Equivalent Circuit. It is drawn by connecting Norton’s voltage I_{N}, Norton’s resistance R_{N} and load resistor in series, as shown below:
From this circuit, the current through the load R_{L} = 10 Ω resistor is obtained using current division rule. It is given by,
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.
Power delivered to the load resistance:
To find the maximum power, differentiate the above expression with respect to resistance R_{L} and equate it to zero. Thus,
Thus in this case, the maximum power will be transferred to the load when load resistance is just equal to internal resistance of the battery.
Results of Maximum Power Transfer:
Note: Maximum power transfer condition results in 50 percent efficiency in Thevenin equivalent, however much lower efficiency in the original circuit.
Example: The maximum power drawn by the load R_{L} in the below circuit will be:
Calculation:
Here R_{th}= 5 ohm and V_{th}= 10 V and R_{L}= 5 ohm.
So Thevenin equivalent would be:
So, power across load can be calculated by calculating current I across R_{L}.
I = Vth/Req
R_{eq }= 5 + 5
= 10 ohm
I = 10/10
I = 1 A.
So, power across R_{L} = I^{2}R_{L}
= 1 × 5
= 5 W
Q.1. What is the superposition theorem?
Superposition theorem is a circuit analysis theorem that is used to solve the network where two or more sources are present and connected.
Q.2. Is the superposition theorem valid for AC circuits?
The superposition theorem is valid for AC circuits.
Q.3. Is the superposition theorem applicable to power?
The requisite of linearity indicates that the superposition theorem is only applicable to determine voltage and current, but not power. Power dissipation is a nonlinear function that does not algebraically add to an accurate total when only one source is considered at a time.
Q.4. Can the superposition theorem be applied to nonlinear circuits?
No, the superposition theorem can only be applied to nonlinear circuits.
Q.5. Why do we use the superposition theorem?
The superposition theorem is very important in circuit analysis because it converts a complex circuit into a Norton or Thevenin equivalent circuit.
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23 videos63 docs60 tests
