What are the network theorems?
The techniques of nodal and mesh analysis extremely powerful methods. However, both require that we develop a complete set of equations to describe a particular circuit as a general rule, even if only one current, voltage, or power quantity is of interest. In this chapter, we investigate several different techniques for isolating specific parts of a circuit in order to simplify the analysis. We shall learn some of the circuit theorems which are used to reduce a complex circuit into a simple equivalent circuit. This includes Thevenin theorem and Norton theorem. These theorems are applicable to linear circuit, so we first discuss the concept of circuit linearity.
All the network theorems are briefly discussed below.
1. Superposition Theorem
The response in any element of a linear, bilateral RLC network containing more Than one independent voltage or current source is the algebraic sum of responses produced by the independent source when each of them is acting alone with
• All other independent voltäge sources are short circuited (S.C.).
• All other independent current sources are open eircuited (O.C.).
• All dependent voltage and current sources remain as they are and therefore, they are neither S.C. nor O.C.
Note:
The theorem is not applicable to the network containing
(a) Non linear elements.
(b) Unilateral elements such as diode or BJT.
The theorem is not applicable to power since it is a non linear parameter. The theorem is also applicable for circuit having initial condition.
2. Thevenin’s Theorem
A linear active RLC network which contains one or more independent or dependent voltage or current sources can be replaced by a single voltage Source Voc in series with equivalent impedance Zeo ,
Voc -→ Open circuit voltage between a and b (when I= 0).
Zeg» Equivalent impedance between a and b, when
(a) All independent sources are replaced by their internal impedances.
(b) All dependent voltage and current sources remain as they are.
Note: Theorem is not applicable to the network containing:
• Non linear element.
• Unilateral element.
3. Norton’s Theorem
A linear, active RLC network which contains one or more independent or dependent voltage or current sources can be replaced by a single current source Isc in shunt with equivalent impedance Zeq
Isc – Short circuit current between a and b(when V= 0)
Zeq- → Same as that of Thevenin’s theorem
4. Maximum Power Transfer Theorem
In any linear bilateral network, the power transferred to the load will be maximum when the load resistance is equal to Thevenin’s equivalent resistance.
Note:
• Efficiency at maximum power transfer is 50%,.
• If Z (source resistance) is varied then maximum power is transferred to the load if Z, = 0.
5. Tellegen’s Theorem
In any network, the sum of instantaneous power consumed by various elements of the branches is always equal to zero. Total power supplied by different voltage sources is equal to total power consumed by various passive elements in various branches of the network.
where, b → Number of branches
Note: The theorem is valid for any type of network as long as KVL and KCL equations are valid.
6. Millman’s Theorem
When number of voltage sources (E1, E2, E3) are in parallel with internal resistances (1/Y1, 1/Y2 ,1/Y3) the arrangement can be replaced by a Single equivalent voltage source Eeq with an equivalent series resistance 1/Yeq
7. Reciprocity Theorem
In a linear bilateral single source network, the ratio of excitation to the response is constant even when the position of excitation and response are interchanged.
Note:
• The basis of the theorem is the symmetry of impedance or admittance of matrix.
• The theorem is valid for network in which linear and bilateral elements are present.
• The theorem is valid only when single independent voltage or current source is present.
• The initial conditions are assumed to be zero in reciprocity theorem.
8. Compensation Theorem
If impedance ‘Z’ of any branch of a network is changed by ‘δZ’, then the incremental current ‘δl’ in such branch is that which will be produced by a compensating voltage source V = 1&Z introduced in the same branch with polarity opposing the original direction of current I.
9. Substitution theorem
The voltage across the current through any branch of a bilateral network being known, this branch can be replaced by any combination of elements that will make the same voltage across and current through it.