#### Superposition and Thevenins Theorem, Physics tutorial

Superposition Theorem:

The superposition theorem for electrical circuits defines that the response (Voltage or Current) in any branch of a bilateral linear circuit containing more than one independent source equivalents the arithmetical sum of the responses caused by each and every independent source acting alone, whereas all other independent sources are substituted through their internal impedances.

To determine the contribution of each and every individual source, all of the other sources first should be 'turned off' (set to zero) via:

Substituting all other independent voltage sources by a short circuit (thereby eradicating difference of potential. That is, V = 0, internal impedance of ideal voltage source is zero (that is, short circuit)).

Substituting all other independent current sources by an open circuit (thus removing current that is, I = 0, internal impedance of ideal current source is infinite (that is, open circuit).

This method is followed for each and every source in turn, and then the resulting responses are added to find out the true operation of the circuit. The resultant circuit operation is the superposition of different voltage and current sources.

The superposition theorem is extremely significant in circuit analysis. It is mainly utilized in transforming any circuit into its Norton equivalent or Thevenin's equivalent.

Superposition theorem is appropriate to linear networks (that is, time varying or time invariant) comprising of independent sources, linear passive elements Resistors, linear dependent sources, Inductors, Capacitors and linear transformers.

Tellegen's Theorem:

Tellegen's theorem is one of the most influential theorems in network theory. Most of the energy distribution theorems and extreme principles in network theory can be derived from it. This was published in the year 1952 via Bernard Tellegen. Basically, Tellegen's theorem provides a simple relation among magnitudes that please the Kirchhoff's laws of electrical circuit theory.

The Tellegen theorem is appropriate to a multitude of network systems. The fundamental suppositions for the systems are the conservation of flow of extensive quantities (that is, Kirchhoff's current law, KCL) and the uniqueness of the potentials at the network nodes (that is, Kirchhoff's voltage law, KVL). The Tellegen theorem gives a helpful tool to examine complex network systems among them electrical circuits, biological and metabolic networks, pipeline flow networks, and chemical procedure networks.

Assume that an arbitrary lumped network having graph 'G' has 'b' branches and nt nodes. In an electrical network, the branches are two-terminal components and the nodes are the points of interconnection. Assume that to each and every branch of the graph we allocate randomly a branch potential difference Wk and a branch current Fk for k = 1, 2.....b, and assume that they are computed with respect to randomly picked related reference directions. When the branch potential differences W1, W2,......Wb satisfy all the constraints imposed through KVL and when the branch currents F1, F2,.......,Fb satisfy all the constraints required by the KCL, then

k =1Σb WkFk = 0

Tellegen's theorem is very general; it is applicable for any lumped network which encompasses any elements, linear or nonlinear, passive or active, time-varying or time-invariant. The generality is expanded when Wk and Fk are linear operations on the set of potential differences and on the set of branch currents correspondingly, as linear operations do not influence KVL and KCL. For example, the linear operation might be the average or the Laplace transform. The other extension is when the set of potential differences Wk is from one network and the set of currents Fk is from a completely different network, so long as the two networks have the similar topology (similar incidence matrix). This addition of Tellegen's Theorem leads to numerous theorems associating to two-port networks.

Definitions:

Incidence matrix: The nt x nf matrix Aa is termed as node-to-branch incidence matrix for the matrix elements aij being

aij = 1, if flow j leaves node i

aij = -1, if flow j enters node i

aij = 0, if flow j is not incident with node i

A reference or datum node Po is introduced to represent the environment and joined to all dynamic nodes and terminals.

(nt - 1) x nf

The matrix 'A' above, where the row that includes the elements aoj of the reference node Po is removed is termed as reduced incidence matrix.

The conservation laws (that is, KCL) in vector-matrix form:

AF = 0

The exclusivity condition for the potentials (or KVL) in vector-matrix form:

W = ATw

Here, wk is the absolute potentials at the nodes to the reference node Po.

By using the KVL:

WTF = (ATw)TF = (wTA)F = wTAF = 0

Since AF = 0 by KCL therefore,

k=1Σb WKFK = WTF = 0

Applications:

Network analogs have been made for a broad diversity of physical systems, and have proven extremely helpful in analyzing their dynamic behavior. The classical application area for network theory and Tellegen's theorem is electrical circuit theory. This is mostly in use to design filters in signal processing applications.

A more new application of Tellegen's theorem is in the area of chemical and biological methods. The suppositions for electrical circuits (that is, Kirchhoff laws) are generalized for dynamic systems following the laws of irreversible thermodynamics. Topology and structure of reaction networks (that is, reaction mechanisms, metabolic networks) can be examined by employing the Tellegen theorem.

The formulation for Tellegen's theorem of process systems: The other application of Tellegen's theorem is to find out the stability and optimality of complex process systems like chemical plants or oil production systems. The Tellegen theorem can be formulated for procedure systems by employing the process nodes, terminals, flow connections and allowing sinks and sources for production or destruction of widespread quantities.

j=1ΣnP Wj (dZj/dt) = k=nf WkfK + j=1ΣnP wjpj + j=1Σnt wjtj, j = 1,..., np + nt

Here pj are the production terms, tj are the terminal connections.

Thevenin's Theorem:

The Thevenin's theorem for linear electrical networks defines that any combination of voltage sources, current sources and resistors having two terminals is electrically equivalent to the single voltage source 'V' and a single series resistor 'R'. For single frequency AC systems the theorem can as well be applied to general impedances, not just resistors. The theorem was first introduced through German scientist Hermann von Helmholtz in the year 1853, however was then rediscovered in the year1883 via French telegraph engineer Leon Charles Thevenin's (1857 - 1926).

This theorem defines that a circuit of voltage sources and resistors can be transformed into a Thevenin's equivalent that is a simplification method employed in circuit analysis. The Thevenin's equivalent can be employed as a good model for a power supply or battery (having the resistor representing the internal impedance and the source representing the electromotive force). The circuit comprises of an ideal voltage source in series by means of an ideal resistor.

Any black box comprising of only voltage sources, current sources and other resistors can be transformed to a Thevenin's equivalent circuit, including exactly one voltage source and one resistor.

Thevenin's equivalent:

To compute the equivalent circuit, the resistance and voltage are required, so the two equations are needed. Such two equations are generally obtained by utilizing the given steps, however any conditions positioned on the terminals of the circuit must as well work:

Compute the output voltage, VAB, if in open circuit condition (no load resistor-meaning infinite resistance). This is VTh

1) Compute the output current, IAB, if the output terminals are short circuited (that is, load resistance is 0). RTh equals VTh divided by this IAB

The equivalent circuit is a voltage source having voltage VTh in series by a resistance RTh.

2)  Substitute voltage sources by short circuits and current sources by open circuits. Compute the resistance between the terminals A and B. This is RTh

The Thevenin's-equivalent voltage is the voltage at the output terminals of the original circuit. If computing a Thevenin's-equivalent voltage, the voltage divider principle is frequently helpful, by declaring one terminal to be Vout and the other terminal to be at the ground point.

The Thevenin's-equivalent resistance is the resistance evaluated across the points A and B 'looking back' into the circuit. It is significant to first change all voltage and current-sources by their internal resistances. For an ideal voltage source, this signifies replace the voltage source by a short circuit. For an ideal current source, this signifies put back the current source by means of an open circuit. Resistance can then be computed across the terminals by using the formulae for parallel and series circuits. This process is applicable only for circuits having independent sources. When there are dependent sources in the circuit, the other method should be employed such as joining a test source across A and B and computing the voltage (notice that R1 is not taken into consideration, as above computations are done in an open circuit condition between A and B, thus no current flows via this portion which that signifies there is no current through R1 and thus no voltage drop all along this portion).

Limitations:

Mostly, if not most circuits are just linear over a certain range of values, therefore the Thevenin's equivalent is valid only in this linear range and might not be valid outside the range.

The Thevenin's equivalent consists of an equivalent I-V characteristic only from the viewpoint of the load.

The power dissipation of the Thevenin equivalent is not essentially similar to the power dissipation of the real system. Though, the power dissipated through an external resistor between the two output terminals is similar though the internal circuit is represented.

Star-Delta Transformation:

In most of the circuit applications, we encounter components joined altogether in one of two manners to form a three-terminal network: the 'Delta', or Δ (as well termed as the 'Pi', or 'π') configuration, and the 'Y' (as well termed as the 'T') configuration.

This is possible to compute the proper values of resistors essential to form one type of network (Δ or Y) which behaves identically to the other type, as examined from the terminal connections alone. That is, when we had two separate resistor networks, one Δ and one Y, each having its resistors hidden from view, with nothing however the three terminals (A, B, and C) exposed for testing, the resistors could be sized for the two networks in such a way that there would be no way to electrically find out one network apart from the other. In another words, equivalent Δ and Y networks behave identically.

There are some equations employed to change one network to the other:

To convert a delta (Δ) to a Wye (Y):

RA = (RAB RAC)/(RAB + RAC + RBC)

RB = (RAB RBC)/(RAB + RAC + RBC)

RC = (RAC RBC)/(RAB + RAC + RBC)

To convert a Wye (Y) to a delta (Δ):

RAB = (RARB + RARC + RBRC)/RC

RBC = (RARB + RARC + RBRC)/RA

RAC = (RARB + RARC + RBRC)/RB

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