Electrical Theorem
1 Introduction to Network Theorems in Electrical Engineering
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 electronics parameters such as voltages, currents, resistance, and so on. These fundamental theorems include the basic theorems like Superposition theorem, Tellegen’s theorem, Norton’s theorem, Maximum power transfer theorem, and Thevenin’s theorems. Another group of network theorems that are mostly used in the circuit analysis process includes the Compensation theorem, Substitution theorem, Reciprocity theorem, Millman’s theorem, and Miller’s theorem.
Network Theorems
All the network theorems are briefly discussed below.
1. Super Position Theorem
Definition :-
Superposition Theorem states that voltage or current through an element of a linear, bilateral network having multiple sources is equivalent to the summation of generated voltage or current across that element, independently by each source present in the network. While at the time of considering a single source all other sources are replaced by their respective internal impedances.
The Superposition theorem is a way to determine the currents and voltages present in a circuit that has multiple sources (considering one source at a time). The superposition theorem states that in a linear network having a number of voltage or current sources and resistances, the current through any branch of the network is the algebraic sum of the currents due to each of the sources when acting independently.
Super Position Theorem
Super Position Theorem
Superposition theorem is used only in linear networks. This theorem is used in both AC and DC circuits wherein it helps to construct Thevenin and Norton equivalent circuit.
In the above figure, the circuit with two voltage sources is divided into two individual circuits according to this theorem’s statement. The individual circuits here make the whole circuit look simpler in easier ways. And, by combining these two circuits again after individual simplification, one can easily find parameters like voltage drop at each resistance, node voltages, currents, etc.
2. Thevenin’s Theorem
Definition :-
Thevenin’s Theorem states that any complicated network across its load terminals can be substituted by a voltage source with one resistance in series. This theorem helps in the study of the variation of current in a particular branch when the resistance of the branch is varied while the remaining network remains the same.
Statement: A linear network consisting of a number of voltage sources and resistances can be replaced by an equivalent network having a single voltage source called Thevenin’s voltage (Vthv) and a single resistance called (Rthv).
Thevenin’s Theorem
Thevenin’s Theorem
The above figure explains how this theorem is applicable for circuit analysis. Thevinens voltage is calculated by the given formula between the terminals A and B by breaking the loop at the terminals A and B. Also, Thevinens resistance or equivalent resistance is calculated by shorting voltage sources and open circuiting current sources as shown in the figure.
This theorem can be applied to both linear and bilateral networks. It is mainly used for measuring the resistance with a Wheatstone bridge.
3. Norton’s Theorem
Definition :-
Norton’s Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load. Just as with Thevenin’s Theorem, the qualification of “linear” is identical to that found in the Superposition Theorem: all underlying equations must be linear (no exponents or roots).
This theorem states that any linear circuit containing several energy sources and resistances can be replaced by a single constant current generator in parallel with a single resistor.
Norton’s Theorem
Norton’s Theorem
This is also the same as that of the Thevenin's theorem, in which we find Thevenin's equivalent voltage and resistance values, but here current equivalent values are determined. The process of finding these values is shown as given in the example within the above figure.
4. Maximum Power Transfer Theorem
Definition :-
Maximum Power Transfer Theorem states that – A resistive load, being connected to a DC network, receives maximum power when the load resistance is equal to the internal resistance known as (Thevenin’s equivalent resistance) of the source network as seen from the load terminals. The Maximum Power Transfer theorem is used to find the load resistance for which there would be the maximum amount of power transfer from the source to the load.
This theorem explains the condition for the maximum power transfer to load under various circuit conditions. The theorem states that the power transfer by a source to a load is maximum in a network when the load resistance is equal to the internal resistance of the source. For AC circuits load impedance should match with the source impedance for maximum power transfer even if the load is operating at different power factors.
Maximum Power Transfer Theorem
For instance, the above figure depicts a circuit diagram wherein a circuit is simplified up to a level of source with internal resistance using Thevenin’s theorem. The power transfer will be maximum when this Thevinens resistance is equal to the load resistance. The Practical application of this theorem includes an audio system wherein the resistance of the speaker must be matched to the audio power amplifier to obtain maximum output.
5. Reciprocity Theorem
Definition :-
The reciprocity theorem states that the current at one point in a circuit due to a voltage at a second point is the same as the current at the second point due to the same voltage at the first. The reciprocity theorem is valid for almost all passive networks.
Reciprocity theorem helps to find the other corresponding solution even without further work, once the circuit is analyzed for one solution. The theorem states that in a linear passive bilateral network, the excitation source and its corresponding response can be interchanged.
Reciprocity Theorem
Reciprocity Theorem
In the above figure, the current in the R3 branch is I3 with a single source Vs. If this source is replaced to the R3 branch and shorting the source at the original location, then the current flowing from the original location I1is the same as that of I3. This is how we can find corresponding solutions for the circuit once the circuit is analyzed with one solution.
6. Compensation Theorem
Definition :-
Compensation Theorem states that in a linear time-invariant network when the resistance (R) of an uncoupled branch, carrying a current (I), is changed by (ΔR), then the currents in all the branches would change and can be obtained by assuming that an ideal voltage source of (VC) has been connected such that VC = I (ΔR) in series with (R + ΔR) when all other sources in the network are replaced by their internal resistances.
Compensation Theorem
In any bilateral active network, if the amount of impedance is changed from the original value to some other value carrying a current of I, then the resulting changes that occur in other branches are same as those that would have been caused by the injection voltage source in the modified branch with a negative sign, i.e., minus of voltage current and changed impedance product. The four figures given above show how this compensation theorem is applicable in analyzing the circuits.
7. Millman’s Theorem
Definition :-
The Millman’s Theorem states that – when a number of voltage sources (V1, V2, V3……… Vn) are in parallel having internal resistance (R1, R2, R3………….Rn) respectively, the arrangement can replace by a single equivalent voltage source V in series with an equivalent series resistance R. In other words; it determines the voltage across the parallel branches of the circuit, which have more than one voltage sources, i.e., reduces the complexity of the electrical circuit.
Millman’s Theorem
This theorem states that when any number of voltage sources with finite internal resistance is operating in parallel can be replaced with a single voltage source with series equivalent impedance. The Equivalent voltage for these parallel sources with internal sources in Millman’s theorem is calculated by the below-given formula, which is shown in the above figure.
8. Tellegen’s theorem
Definition :-
Tellegen’s Theorem states that the summation of power delivered is zero for each branch of any electrical network at any instant of time. It is mainly applicable for designing the filters in signal processing.
Tellegen’s theorem
Tellegen’s Theorem
This theorem is applicable for circuits with a linear or nonlinear, passive, or active and hysteric or non-hysteric networks. It states that the summation of instantaneous power in the circuit with n number of branches is zero.
9. Substitution Theorem
Definition :-
Substitution Theorem states that the voltage across any branch or the current through that branch of a network being known, the branch can be replaced by the combination of various elements that will make the same voltage and current through that branch.
This theorem states that any branch in a network can be substituted by a different branch without disturbing the currents and voltages in the whole network provided the new branch has the same set of terminal voltages and current as of the original branch. The substitution theorem can be used in both linear and nonlinear circuits.
10. Miller’s Theorem
Definition :-
Miller’s theorem states that if an impedance is connected between the input and output nodes in an amplifier, having a reference node N, then this impedance can be replaced by two impedances, one connected between the input and the reference node and the other connected between the output and the reference node.
Miller’s Theorem
Miller’s Theorem
This theorem states that in a linear circuit if a branch exists with impedance Z connected between two nodes with nodal voltages, this branch can be replaced by two branches connecting the corresponding nodes to the ground by two impedances. The application of this theorem is not only an effective tool for creating an equivalent circuit but also a tool for designing modified additional electronic circuits by impedance.
These are all basic network theorems used widely in the electrical or electronic circuit analysis. We hope that you might have got some basic ideas about all these theorems.
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Photo Credits
Super Position Theorem by ee.electrical.world
Thevenin’s Theorem by ee.electrical.world
Norton’s Theorem by ee.electrical.world
Maximum Power Transfer Theorem by ee.electrical.world
Reciprocity Theorem by ee.electrical.world
Tellegen’s & Compensation Theorem by ee.electrical.world
Millman’s Theorem by ee.electrical.world
Miller’s Theorem by ee.electrical.world
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